Vectors are represented by lists with type-encoded lengths. The constructor is
:., which acts like a cons both at the value and type levels, with ()
taking the place of nil. So x:.y:.z:.() is a 3d vector. The library provides
a set of common list-like functions (map, fold, etc) for working with vectors.
Built up from these functions are a small but useful set of linear algebra
operations: matrix multiplication, determinants, solving linear systems,
inverting matrices.

Design

Simplicity :
Beyond the initial complexities of type-level lists and
numbers, I've tried to keep the API simple. There is no vector-space
class, nor a complicated hierarchy of linear/affine/projective
transformations. These can be added on top of the library easily.

Purity :
The library is written in the functional style. For most
functions this does not hinder performance at all, but some I am still
working on (Gaussian elimination) so if this library is a bottleneck you
can easily drop down to C.

Low Dimension :
Although the dimensionality is limited only by what GHC
will handle, the library is meant for 2,3 and 4 dimensions. For general
linear algebra, check out the excellent hmatrix library and blas bindings.

To the point of simplicity, vectors and matrices are instances of Num and
Fractional. All arithmetic is done component-wise and literals construct
uniform vectors and matrices. There are many interesting projects aiming to
overhaul Haskell's number classes, but for now the type of (*) is a -> a ->
a so that's what we're working with. It is easy to incorporate this library
into a more mathematically consistent class hierarchy (provided you can design
one.)

The rule is simple :
If the method is unary, it's a map.
If it's binary, it's a zipWith.

Performance

(:.) is strict in both arguments, but it is also polymorphic, so at runtime
vectors will be realized as linked lists, albeit with less pattern matching.
However the library provides packed representations for 2,3 and 4d vectors of
Ints, Floats and Doubles. Vec3F x y z constructs a packed vector of unboxed
Floats. Functions pack and unpack convert between packed and unpacked
types. When vector operations are bracketed by pack and unpack, GHC can
unfold them into very efficient code. The Storable and UArray instances for
vectors also store them efficiently and generate fast code. Without
optimizations, the code falls back into linked-list mode. The optimizations
depend on inlining, so you may need to increase your unfolding threshold in
certain situations.

GHC Extensions

This library makes heavy use of functional dependencies. I have tried to
tweak things so that they "just work." However, every now and then you will
get incomprehensible error messages, usually about how this isn't an
instance of that. These are how type errors typically manifest, so first
double check to make sure you aren't trying to mix vectors of different
dimension or component types. If you still get these errors, manual type
annotations usually make them go away.